Yes. Each browser vendor runs a push notification backend, where browsers automatically connect to. Each end user browser instance has a unique identifier with that push backend.
As a matter of fact, Mozilla's backend is even open source and written in rust:
EDIT: And to send push notifications to an end user as a service provider, you (in simple terms) receive an encoded version of that URI from the push service, along with an authentication token that authorizes you to send notifications to the push service, which will be forwarded to the user that corresponds to the URI.
As a matter of fact, Mozilla's backend is even open source and written in rust:
https://github.com/mozilla-services/autopush-rs/
EDIT: And to send push notifications to an end user as a service provider, you (in simple terms) receive an encoded version of that URI from the push service, along with an authentication token that authorizes you to send notifications to the push service, which will be forwarded to the user that corresponds to the URI.